Prime ideals of the enveloping algebra of the Euclidean algebra and a classification of its simple weight modules

Abstract

A classification of the simple weight modules is given for the (6-dimensional) Euclidean Lie algebra 𝔢(3) = 𝔰𝔩2⋉V3. As an intermediate step, a classification of all simple modules is given for the centralizer C of the Cartan element H (in the universal enveloping algebra 𝒰 = U(𝔢(3))). Generators and defining relations for the algebra C are found (there are three quadratic relations and one cubic relation). The algebra C is a Noetherian domain of Gelfand-Kirillov dimension 5. Classifications of prime, primitive, completely prime, and maximal ideals are given for the algebra U.